Editing Regular icosahedron (section)

== Related figures ==
The regular icosahedron has a large number of [[stellation]]s, constructed by extending the faces of a regular icosahedron. {{harvtxt|Coxeter|du Val|Flather|Petrie|1938}} in their work, ''[[The Fifty-Nine Icosahedra]]'', identified fifty-nine stellations for the regular icosahedron. The regular icosahedron itself is the zeroth stellation of an icosahedron, and the first stellation has each original face augmented by a low pyramid. The [[final stellation of the icosahedron|final stellation]] includes all of the cells in the icosahedron's stellation diagram, meaning every three intersecting face planes of the icosahedral core intersect either on a vertex of this polyhedron or inside it.<ref>{{multiref
 |{{harvnb|Coxeter|du Val|Flather|Petrie|1938|p=8&ndash;26}}
 |{{harvnb|Coxeter|du Val|Flather|Petrie|1999|p=30–31}}
 |{{harvnb|Wenninger|1971|p=65}}
}}</ref> The [[great dodecahedron]] of [[Kepler–Poinsot polyhedron]] is considered part of subsequent stellation.{{sfn|Wenninger|1971|p=23&ndash;69}}

{{multiple image
 | image1 = Stellation icosahedron A.png
 | image2 = Stellation icosahedron B.png
 | image3 = Stellation icosahedron C.png
 | image4 = Stellation icosahedron Ef1g1.png
 | image5 = Stellation icosahedron G.png
 | image6 = Stellation icosahedron H.png
 | align = center
 | total_width = 800
 | footer = The six stellations of the regular icosahedron according to {{harvtxt|Coxeter|du Val|Flather|Petrie|1938}}: regular icosahedron (zeroth), the first stellation, regular [[compound of five octahedra]], [[excavated dodecahedron]], [[great icosahedron]], and [[final stellation of the icosahedron|final stellation]]. See the [[The Fifty-Nine Icosahedra#Table of the fifty-nine icosahedra|list]] for more.
}}

{{multiple image
 | image1 = Great dodecahedron.png
 | image2 = Truncatedicosahedron.jpg
 | image3 = Triakisicosahedron.jpg
 | image4 = Double diminished icosahedron.png
 | footer = Top left to bottom right: [[great dodecahedron]], [[truncated icosahedron]], [[triakis icosahedron]], and [[edge-contracted icosahedron]]
 | perrow = 2
 | total_width = 300
 | align = right
}}

The [[triakis icosahedron]] is the [[Catalan solid]] constructed by attaching the base of triangular pyramids onto each face of a regular icosahedron, the [[Kleetope]] of an icosahedron.{{sfn|Brigaglia|Palladino|Vaccaro|2018}} The [[truncated icosahedron]] is an [[Archimedean solid]] constructed by truncating the vertices of a regular icosahedron; the resulting polyhedron may be considered as a [[ball (association football)|football]] because of having a pattern of numerous hexagonal and pentagonal faces.<ref>{{multiref
 |{{harvnb|Chancey|O'Brien|1997|p=[https://books.google.com/books?id=wcQIEAAAQBAJ&pg=PA13 13]}}
 |{{harvnb|Kotschick|2006}}
}}</ref>

The [[great dodecahedron]] has other ways to construct from the regular icosahedron. Aside from the stellation, the great dodecahedron can be constructed by [[faceting]] the regular icosahedron, that is, removing the pentagonal faces of the regular icosahedron without removing the vertices or creating a new one; or forming a regular pentagon by each of the five vertices inside of a regular icosahedron, and twelve regular pentagons intersecting each other, making a [[pentagram]] as its [[vertex figure]].<ref>{{multiref
 |{{harvnb|Inchbald|2006}}
 |{{harvnb|Pugh|1976a|p=[https://books.google.com/books?id=IDDxpYQTR7kC&pg=PA85 85]}}
 |{{harvnb|Barnes|2012|p=[https://books.google.com/books?id=7YCUBUd-4BQC&pg=PA46 46]}}
}}</ref>

A [[Johnson solid]] is a polyhedron whose faces are all regular but which is not [[Uniform polyhedron|uniform]]. In other words, they do not include the [[Archimedean solid]]s, the [[Catalan solid]]s, the [[Prism (geometry)|prism]]s, or the [[antiprism]]s. Some Johnson solids can be derived by removing part of a regular icosahedron, a process known as ''diminishment''. They are [[gyroelongated pentagonal pyramid]], [[metabidiminished icosahedron]], and [[tridiminished icosahedron]], which remove one, two, and three pentagonal pyramids from the icosahedron respectively.{{sfn|Berman|1971}}

[[File:Jessen icosahedron with snub icosahedron.png|thumb|Regular icosahedron and its non-convex variant, which differs from Jessen's icosahedron in having different vertex positions and non-right-angled dihedrals]]
Another related shape can be derived by keeping the vertices of a regular icosahedron in their original positions and replacing certain pairs of equilateral triangles with pairs of isosceles triangles. The resulting polyhedron has the non-convex version of the regular icosahedron. Nonetheless, it is occasionally incorrectly known as [[Jessen's icosahedron]] because of the similar visual, of having the same combinatorial structure and symmetry as Jessen's icosahedron;{{efn|1=Incorrect descriptions of Jessen's icosahedron as having the same vertex positions as a regular icosahedron include:
{{harvnb|Wells|1991|p=161}}; [https://web.archive.org/web/20211018124709/https://mathworld.wolfram.com/JessensOrthogonalIcosahedron.html Jessen's Orthogonal Icosahedron on MathWorld] (old version, subsequently fixed).}} the difference is that the non-convex one does not form a tensegrity structure and does not have right-angled dihedrals.{{sfn|Pugh|1976b|p=[https://books.google.com/books?id=McEOfJu3NQAC&pg=PA11 11], [https://books.google.com/books?id=McEOfJu3NQAC&pg=PA26 26]}}

Apart from the construction above, the regular icosahedron can be inscribed in a regular octahedron by placing its twelve vertices on the twelve edges of the octahedron such that they divide each edge in the [[golden section]]. Because the resulting segments are unequal, there are five different ways to do this consistently, so five disjoint icosahedra can be inscribed in each octahedron.{{sfn|Coxeter|du Val|Flather|Petrie|1938|p=4}} Another relation between the two is that they are part of the progressive transformation from the [[cuboctahedron]]'s rigid struts and flexible vertices, known as [[jitterbug transformation]].{{sfn|Verheyen|1989}}

The [[edge-contracted icosahedron]] has a surface like a regular icosahedron but with [[coplanar|some faces lie in the same plane]].{{sfn|Tsuruta|2024|p=[https://books.google.com/books?id=kLckEQAAQBAJ&pg=PA112 112]}}

The regular icosahedron is [[Four-dimensional space#Dimensional analogy|analogous]] to the [[600-cell]], a [[Regular 4-polytope#Regular convex 4-polytopes|regular 4-dimensional polytope]].{{sfn|Barnes|2012|p=[https://books.google.com/books?id=7YCUBUd-4BQC&pg=PA79 79]}} This polytope has six hundred regular tetrahedra as its [[Cell (geometry)|cell]]s.{{sfn|Stillwell|2005|p=[http://books.google.com/books?id=I1QPQic_PxwC&pg=PA173 173]}}